\(\eta _{ab}\)  is the flat space metric on the tangent plane at the spacetime point \(x\) ,\(_{\mu \sigma }^{\nu }\)  is the Christoffel symbol and \(e_{\nu }^{a}\)  is the vierbeinLatin indices will denote tensors in the tangent space basis and Greek indices indices in the space-time manifold. A vierbein relates a quantity with Latin indices to one with Greek indices, e.g. \(H^{abc}=\eta ^{a}_{\mu } \eta ^{b}_{\nu } \eta ^{c}_{\lambda } H^{\mu \nu \lambda }\)  .. The interaction of the sterile neutrino with the background field (REF ) is similar to that found in this microscopic derivation. For the Friedmann-Robertson-Walker (FRW) metric[1]} \(F^{\mu }\)  vanishes. The vierbeins determine the metric through \(g_{\mu \nu }=\eta _{ab}e_{\mu }^{a}e_{\nu }^{b}\) . In the absence of torsion the spin connection \(\omega _{\mu }^{\;ab}\)  is
\(\omega _{\mu }^{\;ab}\equiv e_{\nu }^{a}\left[\partial _{\mu }e^{\nu b}+_{\mu \sigma }^{\nu }e^{\sigma b}\right].\) 
